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Journal of the American Mathematical Society
Journal of the American Mathematical Society
ISSN 1088-6834(online) ISSN 0894-0347(print)

 

Complex Brjuno functions


Authors: Stefano Marmi, Pierre Moussa and Jean-Christophe Yoccoz
Journal: J. Amer. Math. Soc. 14 (2001), 783-841
MSC (2000): Primary 37F50, 11A55, 32A40; Secondary 37F25, 46F15, 20G99
Published electronically: May 30, 2001
MathSciNet review: 1839917
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Abstract:

The Brjuno function arises naturally in the study of analytic small divisors problems in one dimension. It belongs to $\hbox {BMO}({\mathbb{T}}^{1})$ and it is stable under Hölder perturbations. It is related to the size of Siegel disks by various rigorous and conjectural results.

In this work we show how to extend the Brjuno function to a holomorphic function on ${\mathbb{H}}/{\mathbb{Z}}$, the complex Brjuno function. This has an explicit expression in terms of a series of transformed dilogarithms under the action of the modular group. The extension is obtained using a complex analogue of the continued fraction expansion of a real number. Since our method is based on the use of hyperfunctions, it applies to less regular functions than the Brjuno function and it is quite general.

We prove that the harmonic conjugate of the Brjuno function is bounded. Its trace on ${\mathbb{R}}/{\mathbb{Z}}$ is continuous at all irrational points and has a jump of $\pi /q$ at each rational point $p/q\in {\mathbb{Q}}$.


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Additional Information

Stefano Marmi
Affiliation: Dipartimento di Matematica e Informatica, Università di Udine, Via delle Scienze 206, Loc. Rizzi, I-33100 Udine, Italy
Address at time of publication: Scuola Normale Superiore, Classe di Scienze, Piazza dei Cavalieri 7, I-56126 Pisa, Italy
Email: marmi@dimi.uniud.it, marmi@sns.it

Pierre Moussa
Affiliation: Service de Physique Théorique, CEA/Saclay, 91191 Gif-Sur-Yvette, France
Email: moussa@spht.saclay.cea.fr

Jean-Christophe Yoccoz
Affiliation: Collège de France, 3 Rue d’Ulm, F-75005 Paris, France, and Université de Paris-Sud, Mathématiques, Batiment 425, F-91405 Orsay, France
Email: jean-c.yoccoz@college-de-france.fr

DOI: http://dx.doi.org/10.1090/S0894-0347-01-00371-X
PII: S 0894-0347(01)00371-X
Keywords: Small divisors, continued fractions, Bruno functions, complex boundary behaviour, renormalisation, hyperfunctions, modular group, dilogarithm
Received by editor(s): February 16, 2000
Published electronically: May 30, 2001
Additional Notes: This work began during a visit of the first author at the S.Ph.T.–CEA/Saclay and at the Department of Mathematics of Orsay during the academic year 1993–1994. This research has been supported by the CNR, CNRS, INFN, MURST and an EEC grant
Dedicated: This paper is dedicated to Michael R. Herman
Article copyright: © Copyright 2001 American Mathematical Society